## Quasi-Gorensteinness of extended Rees algebras.(English)Zbl 1390.13015

A noetherian ring with a canonical module is called quasi-Gorenstein if it is locally isomorphic to its canonical module. With this definition, a noetherian ring is Gorenstein if and only if it is Cohen-Macaulay and quasi-Gorenstein. While the quasi-Gorenstein property is generally weaker than the Gorenstein property, results of W. Heinzer et al. [J. Pure Appl. Algebra 201, No. 1–3, 264–283 (2005; Zbl 1092.13005)] showed that, in some cases, the quasi-Gorenstein property of an extended Rees algebra $$R[It,t^{-1}]$$ implies that $$[R[It,t^{-1}]$$ is Gorenstein. They also raised the question whether this is true in much more generality. More precisely, if $$(R, \mathfrak{m})$$ is a local Gorenstein ring and $$I$$ is an $$\mathfrak{m}$$-primary ideal, is the extended Rees algebra $$R[It,t^{-1}]$$ Gorenstein if it is quasi-Gorenstein?
In this paper it is proved that the answer is yes for a special class of almost complete intersection ideals. Moreover, if $$R$$ is a polynomial ring of dimension $$d$$, the author shows that the conclusion holds if $$I$$ is a height $$d$$ monomial ideal that has a $$d$$-generated monomial reduction.
The paper contains several results regarding the core of the power ideals $$I^{n}$$ and the $$\mathbf a$$-invariant of a quasi-Gorenstein extended Rees algebra $$R[It,t^{-1}]$$. The author also studies the Gorenstein property of the normalization of $$R[It,t^{-1}]$$ if $$I$$ is a monomial ideal.

### MSC:

 13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)

Zbl 1092.13005
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